ABSTRACT
The purpose of this research is to connect fixed point methods with certain third-order boundary value problems in new and interesting ways. Our strategy involves an analysis of the problem under consideration within closed and bounded sets. We develop sufficient conditions under which the associated mappings will be contractive and invariant in these sets, which generates new advances concerning the existence, uniqueness and approximation of solutions.
Keywords:
third-order boundary value problem; existence; uniqueness and approximation of solutions; contraction mappings; fixed point theorems; closed and bounded sets; two and three-point boundary conditions
1 INTRODUCTION
Consider the following boundary value problem (BVP):
Above: f is a real-valued continuous function; ; and . We define a solution to (1.1), (1.2) to be a real-valued function that has a third-order derivative that is continuous on [a, b] (which we denote by ; and x satisfies (1.1) on [a, b] and the boundary conditions (1.2). Observe that if then (1.2) contains information involving three points; while if then (1.2) expresses data at two points.
There are a number of special cases of (1.1) that have been of interest, especially in the modelling of fluid flow. For example, in the context of mixed-convection boundary-layer flow in a saturated porous medium, the form
was derived in 66. J. Merkin. On dual solutions occurring in mixed convection in a porous medium. Journal of Engineering Mathematics, 20 (1986), 171-179. doi:10.1007/BF00042775.
https://doi.org/10.1007/BF00042775...
),(77. J.H. Merkin. Mixed convection boundary layer flow on a vertical surface in a saturated porous medium. J. Engrg. Math., 14(4) (1980), 301-313. doi:10.1007/BF00052913.
https://doi.org/10.1007/BF00052913...
from the equations of motion. Here x is an unknown part of a stream function that is derived from the governing equations and is to be determined.
Another special case of (1.1) is the classic form
which arises in the study of laminar flow against an infinite plate, see [99. H. Rouse. “Advanced mechanics of fluids”, volume 12. R. E. Krieger Pub. Co, Huntington, N.Y (1976), 444 pp., pp.232-233]. Roughly speaking, in the above notation, the function x depends on distance and is part of a stream function that is related to the velocity field of a fluid experiencing two-dimensional laminar flow against an infinite plate, where a stagnation point is present.
One of the challenges regarding the mathematical analysis of equations (1.3) and (1.4) is the fact that they do not satisfy a Lipschitz condition on unbounded sets and thus they do not satisfy the conditions that are associated with Lipschitz-type existence and uniqueness theorems in unbounded domains.
The right-hand boundary conditions in (1.2) are interesting, for example, due to their ability to approximate important forms. Firstly, if η is fixed and close to b with and , then the three-point boundary condition may be considered as approximating the boundary condition
Secondly, if and then for large values of the boundary conditions (1.2) can be thought of as forming an approximation to the conditions that arise in the modelling associated with (1.3) and (1.4), namely
The literature on third-order BVPs is significant in volume and scope and thus a full review is beyond the focus of this article, however, recent works regarding the existence, uniqueness and approximation of solutions to third-order BVPs that are connected with the ideas in our present article are 22. S.S. Almuthaybiri & C.C. Tisdell . Sharper Existence and Uniqueness Results for Solutions to Third-Order Boundary Value Problems. Math. Model. Anal., 25(3) (2020), 409-420. doi:10.3846/mma.2020.11043.
https://doi.org/10.3846/mma.2020.11043...
), (1111. S. Smirnov. Green’s function and existence of a unique solution for a third-order three-point boundary value problem. Math. Model. Anal., 24(2) (2019), 171-178. doi:10.3846/mma.2019.012.
https://doi.org/10.3846/mma.2019.012...
. Therein, a priori bounds were developed for the associated Green’s function and then applied to ensure the BVP
admitted a unique solution via contractive maps.
Our particular interest in the existence, uniqueness and approximation of solutions to (1.1), (1.2) is motivated by the recent work in 55. A.L.M. Martinez, M.R.A. Ferreira & E.V. Castelani. Theoretical and numerical aspects of a thirdorder three-point nonhomogeneous boundary value problem. TEMA Tend. Mat. Apl. Comput., 20(3) (2019), 417-427. doi:10.5540/tema.2019.020.03.417.
https://doi.org/10.5540/tema.2019.020.03...
. Therein, the authors skillfully developed a set of existence results for positive solutions to (1.1), (1.2) by drawing on ideas from fixed point theory, such as Krasnoselskii’s Theorem and the Leray-Schauder Alternative. A numerical method was also introduced in 55. A.L.M. Martinez, M.R.A. Ferreira & E.V. Castelani. Theoretical and numerical aspects of a thirdorder three-point nonhomogeneous boundary value problem. TEMA Tend. Mat. Apl. Comput., 20(3) (2019), 417-427. doi:10.5540/tema.2019.020.03.417.
https://doi.org/10.5540/tema.2019.020.03...
that employed Picard iterations 1212. C.C. Tisdell . On Picard’s iteration method to solve differential equations and a pedagogical space for otherness. Internat. J. Math. Ed. Sci. Tech., 50(5) (2019), 788-799. doi:10.1080/0020739X.2018.1507051.
https://doi.org/10.1080/0020739X.2018.15...
.
The aim of the present work is to advance the current state of knowledge on (1.1), (1.2) in a way that complements and extends the work in 22. S.S. Almuthaybiri & C.C. Tisdell . Sharper Existence and Uniqueness Results for Solutions to Third-Order Boundary Value Problems. Math. Model. Anal., 25(3) (2020), 409-420. doi:10.3846/mma.2020.11043.
https://doi.org/10.3846/mma.2020.11043...
), (55. A.L.M. Martinez, M.R.A. Ferreira & E.V. Castelani. Theoretical and numerical aspects of a thirdorder three-point nonhomogeneous boundary value problem. TEMA Tend. Mat. Apl. Comput., 20(3) (2019), 417-427. doi:10.5540/tema.2019.020.03.417.
https://doi.org/10.5540/tema.2019.020.03...
), (1111. S. Smirnov. Green’s function and existence of a unique solution for a third-order three-point boundary value problem. Math. Model. Anal., 24(2) (2019), 171-178. doi:10.3846/mma.2019.012.
https://doi.org/10.3846/mma.2019.012...
. A key strategy involves restricting our mathematical analysis to within closed and bounded sets, successfully navigating the aforementioned challenge of global Lipschitz-ness in the process.
We introduce some notation and other componentry associated with our work in Section 2. In Section 3 we establish new estimates on the integrals of derivatives of various Green’s functions, including “sharp” estimates. These new estimates are applied to (1.1), (1.2) in Section 4 via fixed point theorems to ensure the existence and uniqueness of solutions under sufficient conditions. In addition, we establish some constructive results regarding the approximation of solutions through the use of Picard iterations.
2 NOTATION AND OTHER COMPONENTRY
The following two results on fixed points of contractive operators will form important components of our methods.
Theorem 1 (Banach,33. S. Banach. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math., 3 (1922), 133-181.).Let X be a nonempty set and let d be a metric on X such that (X, d) forms a complete metric space. If the mappingsatisfies
then there is a unique such that .
Theorem 2 (Rus,1010. I.A. Rus. On a fixed point theorem of Maia. Studia Univ. Babeş-Bolyai Math., 22 (1977), 40-42.).Let X be a nonempty set and let d and δ be two metrics on X such that (X, d) forms a complete metric space. If the mappingis continuous with respect to d on X and:
then there is a unique such that .
For a recent discussion and comparison of the above theorems, see 22. S.S. Almuthaybiri & C.C. Tisdell . Sharper Existence and Uniqueness Results for Solutions to Third-Order Boundary Value Problems. Math. Model. Anal., 25(3) (2020), 409-420. doi:10.3846/mma.2020.11043.
https://doi.org/10.3846/mma.2020.11043...
.
Herein, we will be concerned with the following notation, sets and metrics. Denote the set of realvalued functions that are continuous on [a, b] by , and for and consider the following metrics:
Now consider the set of real-valued functions that are defined on [a, b] that possess second-order derivatives that are continuous therein. We denote this space by . For functions we furnish the following metrics from d∞ and δ p:
The non-negative constants W i and L i will be appropriately defined in the statements or proofs of our main results.
3 ESTABLISHING ESTIMATES: INTEGRALS OF GREEN’S FUNCTIONS
In this section we establish a range of estimates for integrals involving various Green’s functions from the BVP (1.1), (1.2). These results are of interest in their own right, and we will rely on them to help establish our existence, uniqueness and approximation theorems for solutions to (1.1), (1.2).
By employing the procedure in 55. A.L.M. Martinez, M.R.A. Ferreira & E.V. Castelani. Theoretical and numerical aspects of a thirdorder three-point nonhomogeneous boundary value problem. TEMA Tend. Mat. Apl. Comput., 20(3) (2019), 417-427. doi:10.5540/tema.2019.020.03.417.
https://doi.org/10.5540/tema.2019.020.03...
, it can be shown that the BVP (1.1), (1.2) can be equivalently reformulated as the integral equation
where
and so the partial derivative of G(t, s) with respect to t evaluated at is
We observe the following properties of the Green’s function G(t, s) defined in (3.2).
Theorem 1.The function G(t, s) in (3.2) satisfieson.
Proof. The case is straightforward due the product of non-negative functions therein.
We prove the case via simple inequalities. If we expand the numerator of G and complete the square in t then we obtain
since . □
Let us now develop a bound on the integral of G.
Theorem 2.The function G(t, s) in (3.2) satisfies
Inequality (3.4) is sharp in the sense that it is the best inequality possible.
Proof. For all we have
Now, an application of basic calculus shows that the above expression achieves its maximum value on [a, b] at which gives
□
Let us now develop a bound related to the integral of G t .
Theorem 3.The function G(t, s) in (3.2) satisfiesfor alland
Inequality (3.5) is sharp in the sense that it is the best inequality possible.
Proof. The non-negativity of G t follows immediately from its definition because G t involves the product of non-negative functions.
For all we have
An alternative to the above approach is to swap integration and differentiation to obtain
Now, basic calculus reveals that the above expression achieves its maximum value on [a, b] at which gives
□
Let us now develop a bound on the integral of .
Theorem 4.The functionin (3.3) satisfies
Proof. We have
Thus we have constructed (3.6). □
Let us now develop a bound related to the integral of G tt .
Theorem 5.The function G(t, s) in (3.2) satisfies
Inequality (3.7) is sharp in the sense that it is the best inequality possible.
Proof. For all we have
Once again, this is non-negative, concave up function and its maximum value on [a, b] is achieved at the end points of [a, b]. Its value at and are both and thus we have established (3.7). □
Combining the aforementioned theorems, we have the following results on bounds that will be particularly helpful in the sections that follow.
Theorem 6. For all we have
Proof. The inequality follows from Theorem 2 and Theorem 4. □
Theorem 7. For all we have
Proof. The inequality follows from Theorem 3 and Theorem 4. □
Theorem 8. For all we have
Proof. The inequality follows from Theorem 4 and Theorem 5. □
4 EXISTENCE, UNIQUENESS AND APPROXIMATION
We now construct a range of results that guarantee the existence, uniqueness and approximation of solutions to (1.1), (1.2). Our strategy employs an analysis within closed and bounded sets, the metrics in Section 2, the bounds formed in Section 3, and through fixed point approaches.
So that we do not rely on long and complicated expressions, we define the following constants to simplify our bounds from Section 3. The following notation will be used in the statement and proof of our main results:
4.1 Banach Fixed Point Approach
We can now draw on the results of Sections 2 and 3 to generate new results concerning the existence, uniqueness and approximation of solutions to the BVP (1.1), (1.2) via Banach’s fixed point theorem.
Theorem 1. Let be continuous and uniformly bounded by on the “block”
whereis a constant and each βiis defined in (4.1) with. For, let L i be non-negative constants (not all zero) such that
If
then the BVP (1.1), (1.2) has a unique solution insuch that
Proof. Consider the pair , where the constants W i in our d in (2.4) are chosen to form
(that is, and ). Our pair forms a complete metric space. Now, for the constant in the definition of B, consider the following ball defined via
where
Since ℬ R is a closed subspace of , the pair (ℬ R , d) forms a complete metric space.
Consider the operator defined by
In view of (3.1) we wish to show that there exists a unique such that
Every such solution will also lie in as can be directly shown by differentiating (3.1) and confirming the continuity.
To establish the existence and uniqueness to , we show that the conditions of Theorem 1 hold with .
Let us show . For and , consider
where we have applied Theorem 6. Thus we have
Similarly,
where we have applied Theorem 7. Then, we have
In addition, via similar arguments, we obtain
by drawing on Theorem 8. Consequently, we have
Thus, for all we have
where the final inequality holds by assumption. Thus, for all we have so that .
Let us now show that T is contractive on ℬ R with respect to d. For and , consider
where we have applied (3.4) and (4.2).
Similarly, we can show
Thus, for all we have
Due to our assumption (4.3) we see that T is a contractive map on ℬ R . Thus all of the conditions of Theorem 1 hold with . We conclude that the operator T has a unique fixed point in . This solution is also in and we have equivalently shown that the BVP (1.1), (1.2) has a unique solution. □
The following varitation of Theorem 1 is now established.
Theorem 2. Let be continuous and uniformly bounded by on the “block”
whereis a constant and each βiis defined in (4.1) with. For, let L i be non-negative constants (not all zero) such that
If
then the BVP (1.1), (1.2) has a unique solution insuch that
Proof. The proof follows similar ideas to that of the proof of Theorem 1 and so is only summarized. Consider the pair where now the constants W i in our d in (2.4) are chosen to form
(that is, and ). For the constant in the definition of C, consider the following ball defined via
Since 𝒞R is a closed subspace of , the pair (??R , d) forms a complete metric space.
Following the same type of arguments as in the proof of Theorem 1, for all we have
where the final inequality holds by assumption. Thus, for all we have so that . Furthermore, (4.7) and (4.8) guarantee that T is contractive on 𝒞R .
The existence and uniqueness now follows from Theorem 1. □
Via similar arguments, we have the following result.
Theorem 3. Let be continuous and uniformly bounded by on the “block”
whereis a constant and each βiis defined in (4.1) with. For, let L i be non-negative constants (not all zero) such that
If
then the BVP (1.1), (1.2) has a unique solution insuch that
Proof. Once again, the proof follows similar ideas to that of the proof of Theorem 1 and so we provide just an outline of the ideas. Consider the pair where now the constants W i in our d in (2.4) are chosen to form
(that is, and ). For the constant in the definition of D, consider the following ball defined via
Since 𝒟R is a closed subspace of , the pair (𝒟R , d) forms a complete metric space.
Following the same type of arguments as in the proof of Theorem 1, for all we have
where the final inequality holds by assumption. Thus, for all we have so that . Furthermore, (4.9) and (4.10) guarantee that T is contractive on 𝒟R .
The existence and uniqueness now follows from Theorem 1. □
Let us now form some results that approximate solutions to the unique solution x of the BVP (1.1), (1.2) within certain sets. The ideas follow from Theorem 1 holding for the operator T therein, see [(13), Theorem 1.A].
Remark 4.Let the conditions of Theorem 1, Theorem 2 or Theorem 3 hold. If we recursively define a sequence of approximationson [a, b] via
then, for each of the corresponding metrics defined in the proofs of Theorem 1, Theorem 2 and Theorem 3:
-
the sequence xnconverges to the solution x of (1.1), (1.2) with respect to the d metric and the rate of convergence is given by
-
for each n, an a priori estimate on the error is
-
for each n, an a posteriori estimate on the error is
4.2 Fixed Point Approach with Two Metrics
Once again, so that we can avoid the repeated use of complicated expressions, let us define the following constants with the aim of simplifying certain notation. Let p > 1 and q > 1 be constants such that . Define
and
We will draw on the following relationship between the two metrics δ and d in the proof of our main result.
Theorem 5. For we have
Proof. See [11. S.S. Almuthaybiri & C.C. Tisdell. Existence and uniqueness of solutions to third-order boundary value problems: analysis in closed and bounded sets. Differ. Equ. Appl., 12(12) (2020), 291-312. doi:10.7153/dea-2020-12-19.
https://doi.org/10.7153/dea-2020-12-19...
, Theorem 13] for a proof. □
Let us now state and prove our results on existence and uniqueness of solutions to (1.1), (1.2) where we employ two metrics under Rus’s theorem.
Theorem 6. Let be continuous and uniformly bounded by on the “block” B defined in Theorem 1 with . For , let L i be non-negative constants (not all zero) such that
If there are constants and with such that
where each of the γi are defined in (4.12), then the BVP (1.1), (1.2) has a unique solution insuch that
Proof. Define ℬR , d and T as in the proof of Theorem 1. We want to show that there exists a unique such that
Such a solution will also lie in as can be directly shown by differentiating (3.1) and confirming the continuity.
To establish the existence and uniqueness to , we show that the conditions of Theorem 2 hold.
The pair (ℬR , d) forms a complete metric space. Following the same type of arguments as in the proof of Theorem 1 it can be shown that the condition , ensures .
In addition, consider the metric δ in (2.5) on ℬR where and the L i come from (4.14).
For and , consider
Above, we have used (4.14) and Hoelder’s inequality 44. O. Hoelder. Ueber einen Mittelwertsatz (German). J Goett. Nachr., (1889), 38-47.), (88. L.J. Rogers. An extension of a certain theorem in inequalities. Messenger of Math., 17 (1888), 145- 150. to obtain (4.16). Similar calculations lead us to
Combining the above inequalities we obtain
where
Thus, the inequality (2.2) of Theorem 2 holds.
Furthermore, T is continuous on ℬR with respect to the d metric as can be shown from the following arguments. For all we may apply (4.13) from Theorem 5 to (4.17) to obtain
Thus, given any we can choose
so that whenever . Hence T is continuous on ℬR with respect to the d metric.
Finally, we show that T is contractive on ℬR with respect to the δ metric, that is, the inequality (2.3) in Theorem 2 holds. From (4.16) and the associated discussion, for each and we have
and so we obtain
From our assumption (4.15), we thus have
for some and all .
Thus, Theorem 2 is applicable and the operator T has a unique fixed point in ℬR . This solution is also in and we have equivalently shown that the BVP (1.1), (1.2) has a unique solution. □
Similarly, we have the following two results which we state without proof due to concerns of brevity and repetition. The proofs follow similar lines to that of the proof of Theorem 6.
Theorem 7. Let be continuous and uniformly bounded by on the “block” C defined in Theorem 2 with . For , let L i be non-negative constants (not all zero) such that
If there are constants and with such that
where each of the γiare defined in (4.12), then the BVP (1.1), (1.2) has a unique solution insuch that
Theorem 8. Let be continuous and uniformly bounded by on the “block” D defined in Theorem 3 with . For , let L i be non-negative constants (not all zero) such that
If there are constants and with such that
where each of the γi are defined in (4.12), then the BVP (1.1), (1.2) has a unique (nontrivial) solution insuch that
Remark 9. Let . Each of the invariance conditions can be replaced with
in our existence theorems herein and their conclusions will still hold. To see this, for example, we show that . For all we have
Thus we see that under this condition we ensure that . The other cases may be shown in similar ways.
5 EXAMPLE AND REMARK
Let us probe an example to show how our new results can be applied to concrete problems.
Example 5.1. The nonlinear differential equation (1.3) subject to
has a unique solution such that
where
Proof. We show that the conditions of Theorem 1 are satisfied with and is to be fixed a little later.
We have
Also, on [0, 1], the function ϕ from the proof of Theorem 1 satisfies
Now, the partial derivatives of are continuous and uniformly bounded on B, namely
The value of these bounds can be used as our Lipschitz constants:
The bound M on our f can be calculated via
Thus
and
Now, we see that for the choicewe haveand.
Thus, we see that all of the conditions of Theorem 1 are satisfied and its conclusion holds for this example. □
Remark 1.We remark that the contraction condition (4.15) can be applied to problems where (4.3) does not necessarily apply and it is not difficult to construct a f to illustrate this (see, for example,22. S.S. Almuthaybiri & C.C. Tisdell . Sharper Existence and Uniqueness Results for Solutions to Third-Order Boundary Value Problems. Math. Model. Anal., 25(3) (2020), 409-420. doi:10.3846/mma.2020.11043.
https://doi.org/10.3846/mma.2020.11043...
).
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» https://doi.org/10.7153/dea-2020-12-19 -
2S.S. Almuthaybiri & C.C. Tisdell . Sharper Existence and Uniqueness Results for Solutions to Third-Order Boundary Value Problems. Math. Model. Anal., 25(3) (2020), 409-420. doi:10.3846/mma.2020.11043.
» https://doi.org/10.3846/mma.2020.11043 -
3S. Banach. Sur les opérations dans les ensembles abstraits et leur application aux équations intégrales. Fund. Math., 3 (1922), 133-181.
-
4O. Hoelder. Ueber einen Mittelwertsatz (German). J Goett. Nachr., (1889), 38-47.
-
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» https://doi.org/10.5540/tema.2019.020.03.417 -
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» https://doi.org/10.1007/BF00042775 -
7J.H. Merkin. Mixed convection boundary layer flow on a vertical surface in a saturated porous medium. J. Engrg. Math., 14(4) (1980), 301-313. doi:10.1007/BF00052913.
» https://doi.org/10.1007/BF00052913 -
8L.J. Rogers. An extension of a certain theorem in inequalities. Messenger of Math., 17 (1888), 145- 150.
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9H. Rouse. “Advanced mechanics of fluids”, volume 12. R. E. Krieger Pub. Co, Huntington, N.Y (1976), 444 pp.
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11S. Smirnov. Green’s function and existence of a unique solution for a third-order three-point boundary value problem. Math. Model. Anal., 24(2) (2019), 171-178. doi:10.3846/mma.2019.012.
» https://doi.org/10.3846/mma.2019.012 -
12C.C. Tisdell . On Picard’s iteration method to solve differential equations and a pedagogical space for otherness. Internat. J. Math. Ed. Sci. Tech., 50(5) (2019), 788-799. doi:10.1080/0020739X.2018.1507051.
» https://doi.org/10.1080/0020739X.2018.1507051 -
13E. Zeidler. “Nonlinear functional analysis and its applications. I”. Springer-Verlag, New York (1986), xxi+897 pp. doi:10.1007/978-1-4612-4838-5. Fixed-point theorems, Translated from the German by Peter R. Wadsack.
» https://doi.org/10.1007/978-1-4612-4838-5
Publication Dates
-
Publication in this collection
02 July 2021 -
Date of issue
May-Aug 2021
History
-
Received
27 Oct 2020 -
Accepted
07 Jan 2021